### A generalized beta function and associated probability density.

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We prove the recursive integral formula of class one $M$-Whittaker functions on SL$(n,\mathbb{R})$ conjectured and verified in case of $n=3,4$ by Stade.

2000 Mathematics Subject Classification: 33A65, 33C20.A unifed group-theoretic method of obtaining more general class of generating functions from a given class of improper partial semi-bilateral generating functions involving Laguerre and Gegenbauer polynomials are discussed.

Mathematics Subject Classification 2010: 35M10, 35R11, 26A33, 33C05, 33E12, 33C20.The paper deals with an analog of Tricomi boundary value problem for a partial differential equation of mixed type involving a diffusion equation with the Riemann-Liouville partial fractional derivative and a hyperbolic equation with two degenerate lines. By using the properties of the Gauss hypergeometric function and of the generalized fractional integrals and derivatives with such a function in the kernel, the uniqueness...

The object of the present paper is to illustrate the usefulness, in the theory of analytic functions, of various linear operators which are defined in terms of (for example) fractional derivatives and fractional integrals, Hadamard product or convolution, and so on.

The $q$-derivative operator approach is illustrated by reviewing several typical summation formulae of terminating basic hypergeometric series.